Optimal. Leaf size=28 \[ \frac{\log \left (a+b e^{-c-d x}\right )}{a d}+\frac{x}{a} \]
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Rubi [A] time = 0.022578, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 36, 29, 31} \[ \frac{\log \left (a+b e^{-c-d x}\right )}{a d}+\frac{x}{a} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a+b e^{-c-d x}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)} \, dx,x,e^{-c-d x}\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,e^{-c-d x}\right )}{a d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,e^{-c-d x}\right )}{a d}\\ &=\frac{x}{a}+\frac{\log \left (a+b e^{-c-d x}\right )}{a d}\\ \end{align*}
Mathematica [A] time = 0.0147213, size = 19, normalized size = 0.68 \[ \frac{\log \left (a e^{c+d x}+b\right )}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 41, normalized size = 1.5 \begin{align*} -{\frac{\ln \left ({{\rm e}^{-dx-c}} \right ) }{ad}}+{\frac{\ln \left ( a+b{{\rm e}^{-dx-c}} \right ) }{ad}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10787, size = 46, normalized size = 1.64 \begin{align*} \frac{d x + c}{a d} + \frac{\log \left (b e^{\left (-d x - c\right )} + a\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46414, size = 53, normalized size = 1.89 \begin{align*} \frac{d x + \log \left (b e^{\left (-d x - c\right )} + a\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.133725, size = 19, normalized size = 0.68 \begin{align*} \frac{x}{a} + \frac{\log{\left (\frac{a}{b} + e^{- c - d x} \right )}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27964, size = 47, normalized size = 1.68 \begin{align*} \frac{d x + c}{a d} + \frac{\log \left ({\left | b e^{\left (-d x - c\right )} + a \right |}\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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